Lecture 2 | Quantum Entanglements, Part 1 (Stanford)

this program is brought to you by Stanford University please visit us at stanford.edu I want to start talking about quantum mechanics now we haven't talked about quantum mechanics we've talked about classical physics in a world where what we can call the state space the space of states is discrete we've talked about hopping around from one state to another and how information is counted in bits how with time you can jump from one state to another and so forth that's all classical physics or it's physics based on classical logic classical logic means that the space state is a boolean space so that everybody know what a boolean space is if not all it means is a bunch of points in some space and that each point represents a state and all of the logic is classical logic classical logic that Aristotle would have understood that the Newton would have understood and so forth whereas quantum mechanics makes use of an entirely new kind of logic which is in a very basic way different to to illustrate it I'd like to start with the basic idea first of all of a single quantum bit a qubit we talked about the classical bit which is just the heads tails type distinction here we can I have my prop already this end is different than this end if I point it up that's one state if I point it down that's another state that's the two state system and that's all there is to it nothing in between assume nothing in between that's that's the basic classical bit either up or down with nothing in between ah and of course you're gonna have many bits whoo that's fun you can have many bits to bits three bits four bits whatever and pretty much all the physics anything in physics can be represented or at least approximated by a system of of classical bits anything in classical physics quantum physics the concept of a bit is very very different the concept of a bit of information and that's what we want to talk about today the basic simplest example is the spin of an electron now we don't have to know very much about what spin means does it have something to do with spinning or not this is not very important what is important is that every electron has associated with it a vector now we're going to get into big trouble because there are two things that I'm going to wind up calling vectors one of them is abstract vector spaces which we're going to come to in a little while and the other is vectors in ordinary space a vector I did this I took with me just to illustrate the idea of a vector you know what a vector is in ordinary space it's an object with a length and a direction and I put a nail in the end here to indicate the arrow so that you'll know which way it's pointing it's not pointing at me it's pointing up in the corner of the room there that's your naive elementary physics concept of a vector a thing in ordinary space that has a direction of course it also has components for example this vector here has a component in the horizontal direction it has a component in the vertical direction but altogether it's pointing in in this other direction so that's one concept of a vector a pointer in space with a length in a and a direction now there's another thing we're going to call vectors which is a more abstract concept and there's not necessarily in general will not be referring to things in space it's an abstract mathematics it was a different word for it we're going to be talking about vector spaces I think sometimes when I teach quantum mechanics I'm up a new word for one or the other of them call it a Schecter or something I don't know but it never really works so we have to keep in mind that when I use the word vector sometimes maybe I should just make in the word yeah that's a good idea let's just call this an arrow you know how long I will last before I start calling it a vector not very long okay right now the spin of an electron is a an arrow I was going to say vector I can't help it no I cannot I can't do it I'm going to call it a vector I'm going to call it a vector I can't help myself let's call a spatial vector how about a spatial vector a special vector what means one an ordinary space so if my fist is the electron over here the electron has attached to it a mathematical vector it's not a rod like this but it's a sense of directionality really what it is the direction of twist but just think of it as every electron has attached to it a vector which can point you might think in any direction in space in fact in some sense it can point in any direction in space and in fact that vector can also be thought of as a magnet it can be thought of as a bar magnet with a North Pole and a South Pole so here's the North Pole with a nail the South Pole is over here every electron is a little magnet that's and the mag the amount the strength of the magnet which is measured in some units which I forget in what they called strengthen magnet units is some particular number so that means all electrons there space vectors that go with them the the it's called the magnetic moment the magnetic moments are all the same length they may point in different directions but they're all the same length and we can simply imagine that that they are objects all of the same length of a common length but different directions now let's talk about ordinary classical magnets and I want to talk about the concept of preparing the state and detecting a state we're still thinking very classically for the moment I know everything I explained here was completely with classical logic so here's our electron and it has attached to it an arrow with a North Pole and a South Pole and let's say I want to prepare that electron in a configuration where its mat where its magnetic moment this little vector is pointing this North day it doesn't stay that doesn't stand for its direction it stands for the North Pole of the magnet I want to have the North Pole I want the magnet to be pointed vertically upward I want to prepare it that way how do I do it okay what you do is you simply create a large magnetic field for example you put a you create a magnet this is not a very good pin you create a magnet you can do it as an electromagnet if you like turn on some current make an electromagnet or the North Pole over here a South Pole over here and the electron is supposed to be in the magnetic field and what will happen to that electron what will happen to a little bar magnet well what actually will happen to the little bar magnet if you know anything about how bar magnets move magnetic fields is it will precess it will not just jump up into our right position not unless there's some friction instead what it will do is it will process like this around the magnetic field everything imagine is in a vacuum so there's no friction or anything like that the electron will process around like this that's what that's what a little bar magnet would Daru but sooner or later it will get rid of that processional energy anybody know how it gets rid of the processional energy it radiates some radiation it radiates a little bit of electromagnetic radiation or a lot depending on the circumstances and the size of the magnetic field and so forth it will radiate away its energy and come into the best energetic configuration which is pointing straight up where the North Pole of the magnet is pointing toward the South Pole so where the North Pole of the electron is pointing toward the South Pole of the magnet only which part of I've drawn okay so that's the preparation and if we do it with a very very big magnetic field it will radiate that energy quickly and very quickly come to equilibrium with the North Pole of the electron pointing vertically upward if you like if we don't want to get I was going to use the word entangled but embroiled with the question of where the electron is just imagine in your head that I hold the electron absolutely fixed in position so that the only thing that can do anything interesting is the is the direction of the magnetic field okay so after a small amount of time that that electron will be pointing in the up direction we could if we liked we could prepare the electron in the different configuration by rotating the magnetic poles to something like this south north we could prepare the electron initially so that it's pointing in that direction okay so we start with a electron pointing in some direction now we want to measure we want to do a measurement to find out which direction this is this is a second kind of experiment it's the detection well first was the preparation and then we do some things to the electron whatever it is and next we want to find out we want to detect which direction the electron is pointing in is it pointing in that direction is it pointing in some other direction mainly we could ask what's its angle relative to the magnetic field and to answer that question we do exactly the same thing that we did to detect it to prepare the electron we stick it into the large magnetic field and wait for it to radiate some radiation all right now if this if all the physics were classical and this were really a perfectly classical setup a classical love compass here instead of a quantum mechanical electron then the again the electron would eventually point its way itself up but one measure of the angle would be how much radiation it emits ampel if it was almost perfectly upwards to begin with if we're almost perfectly upward then it only has a little bit of extra energy perfectly upward has the minimum energy when the North Pole is pointing toward the South Pole that's the minimum energy of that little pointer here rotate it away a little bit you give it a little bit of magnetic energy and of course after you've put it into them after you've let it come to rest again it does so by giving off that little bit of energy all right if you were to go to a more extreme situation where you pointed the electron in some horizontally then by the time it got vertical it would emit even more energy finally if you pointed it straight down it would emit the maximum amount of energy in the form of radiation and so you could measure at least the angle relative to the magnetic field by measuring how much radiation comes out right and the answer would be a nice newest function of the angle of the electron so that when the answer when the electron was pointing up no energy comes out this is angle horizontally energy emitted if the angle is zero that's when it's pointing straight up no energy if the angle is downward maximum energy and then as you turn it over all the ways back to zero energy and so forth so you would measure then the same way that you prepare you measure the spin the spin direction of the electron by letting it emit a photon by letting it emit some radiation excuse me and counting up the amount of the radiation now I've told you completely the wrong story for a real electron this is not the way it works this is not the way it works for a genuine electron something else happens and I'll tell you now what it is and it's very weird it really does illustrate the weirdness of the quantum world first of all no matter which way you create the electron in here I won't even try to give a classical picture if I try to give a classical picture I will prejudice the story you do something to the electron and you put it over here whatever you do to the electron one thing you could do to the electron is initially prepare it in some funny angle but whatever you do to the electron you put it over here you turn on the magnetic field and one of two things happens only two things can happen one is it emits no photon no electromagnetic radiation the other is that it emits one quantum one photon of electromagnetic radiation of a very particular frequency it's the particular frequency that corresponds to the energy of jumping down the up whatever that energy is you can emit a photon of just exactly that energy and no other energy so there seem to be it's almost as if the electron had only two possible configurations you jumble it up you don't know what you've done to it you stick it in here and it seems there's only two things that can happen one is that it behaves as if it were pointing up in which case it gives off no photon the other is that it behaves as if it were pointing downward and gives off the photon and those are the only two things that happen that can happen all right but depending on exam that seems a little weird why because supposing you prepared the electron by putting it into a magnetic field that would have the effect of freezing it into a direction let's say at 45 degrees or even better at 90 degrees relative and lowest and reverse horizontal then we would think to ourselves well that electron is now pointing in some other direction now when I stick it into the vertical field you would think that the response would be something different then if it were just plain up or down but no the response is always just one of two responses either it does give off a photon or it doesn't and always of the same energy so it seems that in some sense there are only two states for the electron either it's pointing up or it's pointing down but wait a minute what about if you put the electron into a magnetic field pointing and yet a different direction then you're going to find that there are only two states for the electron either pointing this way or pointing that way something's very confusing how many states does the electron actually have is it only always only up or down or can you make it point in some other direction and if you do make it point in some other direction how come when you stick it into the up/down detector here the detector that detects the direction in the up-down direction that you'll only get two possibilities how come there isn't the continuum of possibilities in between that's the puzzle or that's one of the puzzles of quantum mechanics that it behaves as if there are only two states up or down when you measure it but you can nevertheless prepare that electron pointing in any direction by using a magnetic field in the appropriate direction now the one thing though that is true is if you started with the electron pointing upward let's suppose we created the electron pointing upward by putting it in this very strong magnetic field shut off the magnetic field for a while and then turn it back on the answer is we will get no Photon the electron is pointing up supposing we first create the electron in some other direction pointing along the 45 degree axis here and then we do the vertical experiment we turn on the vertical magnetic field we will either discover that it's up or down but with a probability distribution it will not be definitely up or definitely down we can do this experiment repeatedly over and over and over many many times identically sometimes we will find the photon emitted sometimes we won't the more the electron was initially pointing vertically upward the less the probability is that we will emit that single photon the maximum probability for emitting that photon will be if we were originally pointing down somewhere is in between the probability will be 1/2 for a photon to be emitted in fact that's exactly the case if the original electron were prepared horizontally by a horizontal setup and then put into the vertical field there will be a probability of 1/2 that it emits the photon and a half that it doesn't emit the photon so there's something extraordinary going on here in some sense it seems there are only two configurations every time you measure that electron you will either find it's up or it's down nothing else nothing in between but there are situations where it will be probabilistically if you do the same thing many many times you'll find as a probability distribution and it's that probability distribution R which has some memory of which way you were originally created the photon so a quantum bit is a confusing thing it looks like it has only two states up or down but then you can make a point in some other direction so that it looks like there's other states possible again when you measure it you find only two possibilities it's a confusing entity a quantum bit any questions so far yeah Citian not that nothing my magnet that's attached to an electron is it oh yeah magnetic fields and pies all Chi on you one will try to do this the test and create if you interacted with small objects like other electrons like well it field that's when we're going to start to talk about entangled but let's do one at a time and then we'll come to the two electron system the two electron system is the one that's really interesting because that's where entanglement starts to happen so let's let's let's not try that one yet little steps a little feet yeah quite a question you have to do the whole experiment over and over again prepared if you want to give it experimentally there's no way that with one experiment that you can accumulate a probability distribution okay prepare this right now your turn the sealed off measure again measure again where do you have to prepare another prepare again each time in between okay right right okay yes yes yes you can prepare a lot of electrons at once but but I think the question goes something like this supposing you prepared it this way and then measured it and then measure the same one again and then measure the same in know once you discovered this was up the first time it will be up every time afterwards and so what you will discover in the second experiment the third experiment fourth experiment no photon so once once the experiment has fixed it to be up that's it it has no memory anymore of this all right the other hand is yeah preparing means waiting to see what happens in it when it's it emits you assume yes brightening is doing that again that's that's what I'm cool that's one way of repair that's one way of preparing an electron yeah it's a usual way I'm talking about electron by itself not an atom yeah well I'm talking about an electron perhaps which is in some potential which nails it position down so the Devonian we don't have to worry about it moving we're just we're just concentrating now purely on the spin of the electron so imagine driving a nail through the heart of the electron and nailing it to the blackboard it still has the ability for its spin to move to the one electron Kosta is the electron field to say c'est bien de Monday yes Kosta is it cosmos time okay well okay so is what we're imagine imagining I want to do an experiment so I take an electron which is there originally I don't know how I got it there anyway I like without the magnetic field being there now I want to measure that electron so I very quickly turn on a very large magnetic field how do you do that if it's an electromagnet it's very easy you flip a switch and all of a sudden there's a large electric field sorry magnetic field pointing upward and if that electric APEC's if that magnetic field is large enough it will very quickly cause the electron to radiate okay so I would say no we want to be thinking about turning on the other field to do the measurement or to do the preparation yeah is it possible to prepare facing south and then put it into the kinetic fuel system position then the outcome is getting long that's right if you start you start turning over the whole device you start with South in the bottom where South belongs and north in the top when North belongs then the electron will point down eventually now you remove the magnetic field okay you remove the magnetic field or suddenly reverse just we just turn off the magnet and reverse the polarity you can do that by reversing the current in the electromagnet so that suddenly the South Pole is on top and the North Pole is on the bottom the electron was down in a very short amount of time it'll flip up and give off a photon was that the question and then the probability of that is one yeah the probability that it will give off yes in that circumstance the probability that it will give off a photon is one if on the other hand this weren't perfectly aligned and you're at some caddywhompus angle then the probability would be less than one and it's those probabilities which constitute the experimental experience or somehow I must get a charge out there no it doesn't get electrically charged it has electric charge turning on turning on the electric current going through these magnets you'll have to do a little more you have to put a little more energy into the magnet because the because the electron is there in other words originally the energy comes from the energy that you are that you used to turn on the magnetic field one more in case of the electron actually emits a photon that's the demon sir what happens in that transition well the point yeah I mean that's the language is probably wrong here's what I can tell you if you make the magnetic field sufficiently large the electron will very quickly emit in a photon now the correct answer in general is that the photon will be emitted at a more or less random time in the same way that a radioactive atom will will decay at the case with a half-life okay it decays with a half-life so if you put the electron into the magnetic field in the wrong direction it will with a certain half-life the K the K means give off a photon and write itself in the other direction but if you make the magnetic field large enough that half-life will be very short and so it will very suddenly flip okay during the transition when it's transitioning one of the things we've learned in quantum mechanics is to ask questions only if we know how to make the measurement if you can tell me in detail how to measure when the when the electron emits its photon then we can try to answer it I would say that's one of these questions that is you're not supposed to answer there's a thing called the energy time uncertainty principle and if you know the energy of the electron you you're uncertain about the time but let's let's come to uncertainty principles later it's a good question you have like a million electron you can edit in the intensity of the light tip if you can measure the intensity of the light going off for any given photon emitted at a random bar if you know how many electrons there are MSD white world or capital yes but you can't say with any given on boring we didn't give an electron we Katie what right okay that's a is that more energy system the more that blew the piggy you making magnetic field more energy you put in more quickly it will that it will right absolutely absolutely more quickly in the sense of a smaller half-life right okay so first of all we're dealing with a theory that only has problem that has a probabilistic description and we will see ultimately that there is really probably there's probably no way around the fact that the description has to be probabilistic at least at some level the mathematics of the state space of the of the concept of a state the state is different in quantum mechanics it's clearly different if this were clearly different it's a system with only two states you think but then again it seems to be able to be oriented in any direction the mathematics of quantum mechanics and the states are in particular the states of a quantum bit are not the mathematics of a set of two points they're not the boolean mathematics of a set of two points this thing which describes the mathematics describes a quantum state is a vector space now I use vector in a different sense it is no longer an arrow in ordinary space it is a mathematical abstract vector space whose character you'll come to understand if you persist in the and persist at it a little bit you will begin to understand what this vector space is and how it characterizes the possible configurations or the possible states of an electron and how it can be consistent with this fact here okay I'm going to spend a little bit of time now just doing some abstract mathematics very abstract mathematics and then we'll seek to interpret it and to interpret the state of the electron I'm going to tell you what a linear vector space is what a vector space is a vector space is a collection of objects and here are the rules for an abstract vector space though as I said do not confuse this with the pointer that that is a vector in ordinary space this is just an abstract completely abstract concept first of all the vector is an object and we'll label it like that it's an object with a set of rules a set it's a collection of objects it's a collection of objects with a rule first of all a rule that any vector can be multiplied by a constant to get a new vector this gives a new vector we could call it let's call it a prime so there's a concept of multiplication by a constant now first of all let me tell you that the constants we're going to be talking about are complex numbers not real numbers anybody here not familiar with the concept of a complex number if so raise your hand and be humiliated okay all right now if you really don't know what a complex now it may be that you know it but it by another word now if you really don't know about complex numbers look them up oh because you're going to need them alright a complex remember this this number I I is the square root of -1 now -1 has no number that's through school or there's no number ordinary number whose square whose square is minus 1 so there is no square root of minus 1 any number times itself is always positive right 3 times 3 is 9 minus 3 times minus 3 is also 9 all right but mathematicians many many many years ago invented the abstract idea of an imaginary number they called an imaginary and said let it be that I squared is equal to minus 1 invent the new number okay we know how to use those numbers I'm not going to I'm not going to spend any time at it whoever raised a hand are they familiar with I equals the square root of -1 yes all right now you can have complex numbers complex numbers or what happens if you take two two real numbers ordinary numbers let's call it a and B different a plus I B in other words if you add a real number to an imaginary number you get a complex number that's the whole definition of a complex number you can multiply complex numbers and all you have to remember you use the ordinary rules of arithmetic except you have to remember that I squared is minus 1 so for example if you're going to well as an exercise you can square this what's the square of a plus IB I won't tell you go work it out all right so that's a complex number often character often written is Z sometimes either you can use any letter that you like but it's a complex number with both a real part than an imaginary part another concept is the complex conjugate this is very important you'll need it all right and it's very simple it's a minus IB is the complex conjugate of Z so if you take a real no if you take a complex number and you change the sign of the imaginary part that's called the complex conjugate of the number the complex conjugate of the complex conjugate gives you back the original number that these are important mathematical concepts they're very elementary let me give you an example supposing you multiply Z star by Z what is that that's a or Z times Z star it doesn't matter either way it's a plus I B times a minus I B let's see what we get just ordinary multiplication we open up the brackets here we get a times a which is a squared we get a times IB with a minus sign minus I a B plus I a B from this one times this one and then what about IB times IB what does IB times IB give minus B squared but there's a minus sign here already so it's minus I times I times B and so that gives plus B squared all together we get a squared plus B squared these pieces cancel so that's an example of using common mean that's an example now we're not using it for anything it's just an example of a definition and then using the definition of complex arithmetic to calculate what Z star Z is notice that Z star Z is always real and it's always positive it's always real and positive a squared a and B are real numbers a squared + B squared are both positive so Z star Z is a real number and it's positive it's lot of more or less is the magnitude of the come of the of a complex number all right so first of all for the kind of vector spaces we're going to be thinking about they're called complex vector spaces and one of the rules of one of the operations that exists in the complex vector space is multiplication by a complex number now a complex number includes real numbers incidentally so you can see could be 2 or it could be I or it could be 2 plus I or anything like that there's a notion if there's a vector you may multiply it by a constant and you get a new vector that's the first operation and there's only one other operation that's of interest to us it's adding vectors there's a rule for adding vectors or assume there's a rule for adding vectors so if I have two vectors a and B I can construct an vector called a plus B and that's some new vector which I guess we can call C but C is a different C that appears here this C stood for constant this C just stands for a new vector which is a plus B so every pair of vectors you can add them and I'm going to show you I'm going to show you some less abstract examples but first of all the simplest example of a vector space like this is just the complex numbers themselves the ordinary complex numbers allow you to multiply any complex number by another complex number in particular a constant particular kind of complex number and it allows you to add complex numbers so just complex numbers by themselves or a vector space a complex vector space that's the simplest example but let me show you another example and the other example is something we talked about last time which we called column vectors just represent an abstract quantity by a set of components all the components are themselves complex numbers so for every vector really what that vector is standing for I write equal signs I probably shouldn't write equals mathematicians would have a stroke if I wrote equals their is represented by a column vector the column vector has entries and it's just a table of numbers but it's a table of complex numbers all right the table of complex numbers supposing you want to multiply that column vector here by a constant the rule is very simple multiply every entry by that constant so C times AE is represented by ca1 ca2 ca3 we just multiply all the components by the same number that's multiplication by a constant how do we add two vectors together let's suppose we have two vectors a and B a is represented by the column whose entries are a 1 a 2 a 3 B is represented by B 1 B 2 B 3 well these aren't with the bays and B's are just numbers and when we add them we just get a third column vector whose entries are a 1 plus B 1 a 2 plus B 2 a 3 plus B 3 so column vectors are an example of a vector space and they are the basic example that we will use over and over again as I said mathematicians would have a stroke if I call the vectors that would say the columns represent vectors and you would call these the components or a set of components of a vector okay um that's the idea of an abstract vector space the very strange thing is that the states of a quantum system in particular a quantum bit Oh incidentally you're not restricted to three this is a particular case this is a three dimensional vector space all right we could have a two dimensional vector space that's actually going to be more interesting to us for the moment a two dimensional vector space just has two entries that's the next simplest thing if you like to the numbers the next simplest vector space is a two dimensional vector space and there's a three dimensional vector space and a four dimensional vector space and so forth and they will make sense okay that's you'll get used to this sooner or later you'll you'll start thinking in terms of abstract vector spaces if you pursue this subject there's no way to learn quantum mechanics honestly and correctly without going through this mathematics so it's it's absolutely essential now John if we were talking about the simplest vector space just complex numbers then every complex number has its complex conjugate that means that there is a second vector space which is the space of the complex conjugate numbers right that the concept of complex conjugation is an important concept and it also exists for other vector spaces all you have to do basically is write the complex conjugate of the entries but a rule when you're writing the complex conjugate of a vector I want to write the complex conjugate of the vector AE you draw it the other way this way that stands for the complex conjugate it of the vector a and furthermore you represent it by a row vector the components of the row vector are the complex conjugates a 1 star a 2 star so whenever you see a row vector always think of it as the complex conjugate of the corresponding of a corresponding column vector if I tell you this are a row vector a which is in correspondence with a column vector a II always remember that there's a complex conjugation operation this is the basic idea of a complex vector space written out in components everybody happy with that anybody those no asterisks that the a no usually that's right it that that's right the the notation does not put a star over here but do you remember that the star is implicit in turning the symbol backward now I'll remind you this symbol is called the ket now the symbol is called a bra this symbol is called a ket do I have it right I don't another one of them is a bra one of them is a ket but it doesn't matter it's completely symmetric between the two remember that the complex conjugate of a complex conjugate is the original thing so that's that's the notion of a complex vector space and finally not finally but we want to put the pieces together you can multiply two vectors the product it's called we've discussed this before I realize I'm not getting senile I know we're talked about this before but we need we really need to do it right now there's the concept of the inner product of two vectors the inner product of two vectors the easiest way for me to describe it is in terms of components I don't want to spend a lot of time giving you the most abstract definition are there many mathematicians in the audience plug you ears just just just just plug them up and go all right um they actually mathematicians also use components but um I think anyway the next concept is the concept of the product of two vectors but it's always and it's the inner product of the two vectors and the concept is really the product of a vector with the complex conjugate of another vector it's very much like multiplying a complex number by the complex by a complex number by the complex conjugate of another number could be the same number could be another number so I'm going to give you the rule the rule I'm going to tell you in terms of components okay here's the rule if I want the inner product between the vector B and the vector AE is written like that it's got a bra ket that's a bracket okay that's where the term bra ket came from half a brow half a bracket as a bra the other half of a vector is a cat and again I forget which one is which this is the cat right yeah that's the cat this one's the bra okay good good the rule is you take the two that you take the two vectors the B vector is a row vector and you're right b1 star b2 star that's the representation of the B vector and the representation of the a vector is a 1 a 2 you do not star it and the inner product between them this product has gotten by multiplying the first component of B with the first component of a eb1 star a 1 plus B 2 star a 2 first first row with first sorry first entry here with first entry here plus second entry here second entry here supposing for example okay everybody that's that's the basic concept of the inner product it's just like multiplying two numbers together except we get a term for each for each entry one entry in the two entry for example if we take the inner product of a vector with itself let's take a with AE the inner product of a vector with itself as the form a 1 star a 1 plus a 2 star a 2 notice one thing about this a number of times its complex conjugate is first of all positive and second of all real well it couldn't hardly be positive if it weren't real it's a positive number so the sum so the inner product of a complex vector with itself is always a real positive number it can be thought of as a kind of size of a vector it's actually the square of the size of the vector if you like you can think of it as a square of the size of the vector the length of the vector it measures it measures the magnitude of the vector so this operation of inner product is absolutely essential as we'll see to the interpretation and it's complex vectors like this which represent the states of a quantum bit let me just digress when you stop doing mathematics for a minute and talk about the quantum bit again the quantum bit could be pointing straight up we could invent a symbol for the state for the configuration of the electron pointing up let's call it we could either we could call it up or we could call it a little vector up or we could just call it plus I think I'll just call it plus to indicate that it's pointing upward or we can have an electron which is pointing down now remember when we go to measure the electron we either find that it's up or down and nothing in between and that we can identify with the vector minus just an abstract notation this stands for the state of an electron pointing up this stands for the state of an electron pointing down now in classical physics we would never under any circumstances think of adding these two vectors or multiplying them by numbers we would just say this means electron and up this means electron down in quantum physics the general state of an electron or an electron spin is a vector in a vector space which we could write a plus x plus plus a minus x minus in other words it's a two-dimensional vector space we could also represent it by a plus a minus and he is the rule the rule that's so there are some coefficients that we can add them together with and here's the meaning of those coefficients or at least a partial meaning of those coefficients a plus star a plus is the probability that we find the electron up remember it's positive it's positive and it's real a star times a is positive and it's real and it's the magnitude of the or the square of the magnitude of the coefficient of the up configuration and then there's the down configuration the probability of finding the electron down is a minus times a star minus that's the probability to find it down that's the interpretation or that's part of the interpretation just to give you some orientation where we're going the quantum state of that electron which when we measure it we find with a probabilistic distribution is either up or down and nothing in between that's represented by saying the electron when it's pointing in some funny angle is given by a complex vector complex vector space which can either be represented abstractly like this it can be represented concretely as a column vector and the square or magnitude of the square of the entries here are simply the probabilities to find it up or down now presumably probabilities add to one so one of the rules about quantum mechanics is that the vectors that represent that we have a vector space of some kind an abstract vector space but the vectors which actually represent the physical states of a system are the word is normalized normalized means that the sum of the probabilities is one so a plus star a plus plus a minus star a minus should be set equal to one this is like considering only vectors of unit length that doesn't mean that in the vector space itself there aren't other vectors of other magnitude there are but the physical states of a quantum system have to be normalized which is the same thing as setting the sums of the probabilities to one so normalized vectors in the vector space represent the states of a quantum system now that's a very abstract concept why do we get driven to such very abstract concepts why can't we visualize this the same way we visualize classical physics well it's because we don't have the wiring and we have to rewire ourselves and we have to rewire ourselves to learn to think about the quantum states of a system being an abstract vector space once you get this idea of an abstract vector space and how it fits together with the states of a system then you're flying and you can understand all of quantum mechanics so any questions about this the entries the coefficients or equivalently the components of the vector when you square them or better yet when you multiply them by their own complex conjugate give you the probabilities for the two possibilities yeah that represents a quantum state if a star a a in fact we can say it this way the inner product of a with itself that's given by a star a plus star a plus plus a minus star a - that should be one so legitimate quantum states should have inner product with themselves which is equal to one and that's just a statement if it's that's just a statement that the sums of the probabilities of all the probabilities where is very for all those like column vector a plus B minus o plus and minus are now just to indicate that one of them corresponds to spin up and the other corresponds that has been down and we could have called them eight when we could have given them different names I could have called them a up and a down or I could have called them a1 and a2 they do have they do have they are the coefficients of the ket plus and the ket - okay so let's go back a step back a step what about the ket plus how should we represent that in some basis let's well at the moment I don't want to discuss the the ambiguity of in basis vectors will come to that but for the moment we have a basis the Moriya but you don't know I'm talking about this vector will be represented by one zero okay it'll never be represented by a 1 in the upper place and a zero in the lower place what about the - cat the - cat is represented by a 0 and a 1 this is just some arbitrary correspondence okay now then if I multiply this by a plus the numerical constant a plus then we have to multiply this by a plus we don't have to multiply 0 plus a plus because 0 times anything is 0 if we multiply this by a minus then this becomes a minus here and if we add them together a plus x the plus cat plus a minus times the minus cat we just add these together and we get adding them gives us a plus a minus okay does that answer the question that was asked okay so yes we said certainly the plus and minus notation instead of one and two here I used 1 & 2 they're just labels they're just labels 1 & 2 is just a label to label the two entries I could have used up and down I could have used Joe and Pete I could have used anything I want to label them here I'm using plus and minus the two possibilities for the orientation of the electron of the electron or of the quantum bit I'm labeling them by plus and minus good okay so now we have a a basic concept of a vector space the inner product - Jerry get over to our come here just - okay reminders kept mine is one nice no the - ket is not minus 1 times the plus ket no no no no no no no oh yeah this matter yes yeah you're asking you're asking whether the whether this object makes sense yes you're allowed to take any vector and multiply it by any complex number in particular a complex number is minus 1 in fact but this but this is not it is not the same as - right this has a fist okay let's let's make that clear the - here does not mean that it's - the + vector it's just a label no more than 8 then the then the a 2 means twice a 1 is that the popular altruistic estate so - plus cat and cat both represent state where you have probably one trusted yes that's correct that's correct oh no the line is candid probability want to the highness state wait wait this cat here has probability one in the plus side but so I love minus sign we also have which one per corner if we remove them I - that's right that's right both of these that's right absolutely absolutely we're gonna come to that very good very good very good very good well you're you're pointing out something that I was about to come to in a moment I'll tell it to you right now well okay if I multiply it by any complex number of magnitude one its ass that's right that's correct that's correct so physically we would not distinguish this state from from this state I won't write equals because in the mathematical vector space they're not equal but they represent the same physics however however when I add them a plus and minus a minus are not the same that is a different state all right that is a different state with a minus sign here but we'll come to that we'll come to that it was something I was going to talk about but not yet I want you to digest just the idea that the components of these abstract vectors when squared correspond to probabilities and they make physical sense to tank Jinxy - Ken - the broth off are those two together yes you're gone - is my head so red spots let's so it's coming I think you're asking whether it makes sense to take the cat vector plus and consider what I'll call its conjugate which is the bra the bra vector plus in this case the there's no change because Z well we can accept that you would write it as a row right what about the in a proton is they'll change that's right in this case in this case right the course one is a real number we don't have two complex conjugated okay so here's a here's a little exercise what's the inner product of the Plus vector with itself one one because it's just multiplying 1 0 times 1 0 1 times 1 is 1 so this is 1 what about - - also equal to 1 what about the inner product of plus with - let's just check that for those who don't see it so quickly let's just check it the plus bra is one zero the - ket is 0-1 all right so we multiply them together we get 1 times 0 plus 0 times 1 which is all together 0 what about - plus also 0 what's the word for two vectors whose inner product is 0 both agonal orthogonal they're called orthogonal vectors so the two vectors representing the two distinct states of the electron one up and one down are two orthogonal vectors in a linear vector space of states a linear vector space of states okay that's the basic set up our and the coefficients that appear there they're squared magnitudes are the probabilities for the two configurations the probability for up and for down now as you can see there are more possible physical states than just up and down there are all the linear combinations it's the linear combinations of them that correspond to the electron point there having been prepared in different directions so in other words an arbitrary combination like this with a plus and a minus there when this is not 1 in 0 1 0 and 1 correspond when you consider all the possible complex numbers to the different directions that you might have prepared the electron but still all there is is up and down and the probabilities for up and down so a state for example let me give you an example a vector a vector let's write one in both and both slots that's not allowed why isn't that allowed probably doesn't add to one alright so the probability would add to 2 here 1 squared plus 1 squares do we have to divide it by square root of 2 now each of the probabilities is 1/2 and they add up to 1 parentheses where you're right I make the right to column that's a state with equal probability namely probability 1/2 to find the electron up and now this one has probability 1/2 to find the electron down there are more states than just up and down and in fact this one corresponds to what you would create if you created the electron pointing in the horizontal direction if with a magnetic field in the horizontal direction you froze the electron let's take a break if you froze the electron the place with a magnetic field in the horizontal direction in the horizontal plane one of the possibilities would be this state here which would have half a probability of being up and half a probability of being down half the probability would be up and half the probability being down actually corresponds to some configuration where the electron is lying somewheres in the in the horizontal plane so we have plenty of vectors around with all kinds of complex numbers out of which we can build and we'll see that we can do this we can build these the space of electrons which in some sense of pointing in any direction but whenever we measure them all we get is up or down with probabilities that are governed by these coefficients let's take a break for 7 minutes 7 or 7 minutes and 30 seconds all right now more than one person is probably a little bit confused about the two notions of vectors one is having to do with directionality in ordinary space and one having to do with this abstract concept and this notion of components of vectors labeling upstate and downstate question is what's the connection between the notion of directionality in space and these vectors here now what I want to say first of all is they are not simply related they are related but not simply related for example how many components does a vector have an ordinary vector three on and then and they're real numbers right XY and Z they're all real numbers these vectors have two components and they're complex numbers and that means how many how many real numbers does it take to describe one of them for all right so they're not the same thing but there must be some connection we're not going to do that connection just yet but we will come to it next concept which we already went into a little last time is the concept of a matrix is right okay it's an experimental question from the fact that there were only two possible answers to the way or that when you do this experiment on the electron you either get that it's up or down and nothing in between that tells you that there are two possibilities and it tells you that you should be dealing with a 2-dimensional vector space all two-dimensional complex vector spaces are mathematically the same so they're all the same a two-dimensional complex vector space is a two-dimensional complex vector space there's no there's not more than one of them mathematically so ah so you count up the number of possibilities somebody asked me before if I had to spin one particle instead of a spin 1/2 particle which is what an electron is how many components would there be then there would be three but it's an experimental question how many possibilities are there how many states are there how many how many distinct possibilities are there and in the case of the electron spin in the case of the spin of the electron there are two possibilities and so it's a two dimensional vector space and they're all the same there's no there's no distinction between different two dimensional vector spaces other than the fact that we could consider the real or complex vector spaces okay let's let's move on to the concept to the abstract concept of a linear operator or the concrete concept of a matrix let me tell you where we're going we've talked about States but we haven't talked about the things that you actually measure the things that you measure or that you can measure are called the observables the observables of a system are the things that you can measure and get answers for position of an electron isn't observable but we're not doing anything as complicated as that as that if we were to label the up and the down state as plus and minus we could invent an observable which would have two possible values it could either be plus plus one or minus one and it would be an observable that we could measure if the electrons up we will assign it the number plus one if it's down we'll assign it the number minus one also called an observable anything that we can measure that has a numerical value numerical value means a real number anything that we can measure that when that when the measurement is recorded then it gives rise to a number in numerical number is called an observable so as I said in the case of the electron spin if we measure the if we measure the whether it's up or down we could assign up the number plus one and down the number minus one and in that way have an observable which whose numerical value when we measure it is either plus 1 or minus 1 let me talk about before talking about observables in quantum mechanics let's talk about observables for a minute in classical mechanics including classical theory in classical theory the states of a system are just a set of states which we represented by points how many you have 1 2 3 4 5 6 these could be the six possibilities for throwing a dice a by a single die the 1 2 3 4 5 6 this is and we could assign some numbers to each one of these points we could call we could label this 1 1 2 3 4 5 6 we throw the die and we look at it that's the measurement we throw the die and we look at it and we get an answer the answer is either 1 2 3 4 5 or 6 that measurement is the measure than above an observable and it's an observable which has six possible answers there are other observables that we could concoct for example we could concoct an observable which is zero everywhere except at one value one zero zero zero zero zero now this means that if we flip the die and we get a 1 we assign the number 1 to the observable if we get any other number we assign the value 0 to then it could be observable so there are many observables that you can make and basically they correspond to any functions of these points any functions of these points assign any function to the points function points meaning assign any numbers you like to these points real numbers and then when you flip the die and you see what comes up you say the value of the observable is whatever the numerical value of that observable was when you flip it back there are many many observables that you can think about in this in this simple system but they're all rather trivial I mean they just correspond to to the basic question of whether the die is 1 2 3 4 5 or 6 um it's interesting just not because yes it is interesting to define the observable as a function of which state we're talking about F sub n n here represents which of these states were talking about and F is a function which could be any number for each of these states that's called an observable when you flip a die you look at what you get and you assign the number F the particular state F sub one F sub two of through F sub six that's called an observable and that's a it's a sort of overkill concept in this classical situation here but but let's continue now let us suppose it for one reason or another in classical physics we haven't been very careful in doing our classical physics measurements and so forth and we don't know exactly what state the system is in all we know is some probabilities for example we may have a loaded die a loaded die which is an unfair die like the kind the dirty gamblers use and you flip it and it lands on the table there may be a probability distribution for different values of the one two three four five or six let's label that P sub n also that's the probability the a priori probability let's say that when you flip the die you get the nth state then what is the average if you flip the die many many times and each time you measure F you can flip it many many times you measure F what is the average value of F after many many many identical experiments okay the answer anybody know well the answer for the average value of F or we're assuming of course that the sum of the piece of ends is 1 in other words the sum of the probabilities of 1 the average value of F is the sum of all the possibilities of the probability for the nth power for the nth the configuration times the value that the function has in the nth configuration so what you do is you add up all the configurations weighted according to their probability weight them according to their probability for example if all the proper all the probabilities are equal all right then you just add up the F sub ends if they're all equal incidentally they're not one there the 1 over N because they're 1 over the number 1 over 6 in this case but this is the average over many many let's let's take an example let's do a coin a simple coin or up or down heads or tails let's say there's the probability for heads and we're going to call heads Plus is the probability for heads which is P plus there's the probability for tails which is P minus alright if we have heads we assign the number plus 1 if we have tails we assign the number minus 1 what is the probability for that observable which is either plus 1 or minus 1 it's P plus the probability for plus minus the probability for minus because every time you get a minus you give it a value minus 1 every time in other words every time you get a tail you give it a sauna minus 1 every time you get a head you give it a plus 1 the average is P plus minus P minus the difference between the probability of head and tail is the average of the heads Nisour the tails nests so this basic formula here which is just the most elementary formula of probability of the notion of an average that you average over all the possibilities weighing it with the value of the quantity the observable whose average you're taking and you weigh it according to the probability of that particular configuration and that's the average value of an experiment which consists of many many repeated experiments what's that expected value sometimes called the expected I sometimes called the expectation value sometimes called the average value and and that's what we're going to be interested in we're going to be interested in the average values of things that we can measure the average values of observables now I what I haven't told you is how you represent observables in quantum mechanics and the representation of observables in quantum mechanics is more intricate and complicated than just saying we assign to each state a numerical value that we call the observable value in that state it's a more complicated concept and a more tricky concept the notion of an observable or measurable measurable quantity and it's related not related but is well related related to the concept of a linear operator or a matrix so for our purposes the same thing again the mathematicians will object to me saying a matrix is a linear operator I don't know the other job anybody object my saying a matrix is a linear operator what yes courage to do this example yeah of plus is 1 and F of minus is minus 1 desperate they should value though is you're off if the probabilities are equal yeah well I think probably is equal so is this the expectation value ends up being a number that is that the underlying system is never in that state oh that's true yeah yeah yeah right that's why people don't like calling it the expected value because it is the it is a value which is very definitely not expected so I everybody used to call it the expectation value now that's a little that's a little weaker than saying the expected value then my friend Murray gell-mann started calling it the expected value well it ain't as you point out the expected value is more than that because you have to bark on you really haven't by by N and as grown the sums of the peas are up to one assuming the sums of the peas add up to one okay but when you think of it as dividing by the overall sample size in this case yeah that's a piece of P is 1/2 and 1/2 that means that you the typical person having two and a half kids okay right so there's nothing it's not really be expected you know you have to guess how many kids you're going to have you probably wouldn't get two and a half it's a payoff in the gambling you can actually run up a zero well you said so it's about you get it you put the corresponding point then this is the average value it's by definition the average value okay so that's right so if I'm glad your point to this count that if P + + P - were both 1/2 then the add then the average value would be 0 but 0 is not a possible answer so the average value is not the expected value there is no expected value in this case but it doesn't something you get something at 0 the average value it considered a lot higher villages and self-esteem so again it hasn't it as well as I would a virgin's that's a lot the variance is to Wi-Fi well I think coin we might flip it a fair coin and I flip it I think we'll get 1/2 times heads and F times tails you said something about you see something about temperature there they are yes in the case of composting of course it tells you that you is a 50/50 percent chance no hey Jeff yeah I told you you put your basically completely ignorant if you're completely ignorant there's a 50/50 chance and and if you do the experiment over and over with a fair coin and you average the result the average had better be zero because it can't be can't be positive and it can't be negative because equal balance on the other hand if you have an unfair coin let's say three quarters of the time it comes up heads and one quarter of the times that come the pails then the average is what it's 3/4 minus 1/4 which is 1/2 all right so if you have this unfair coin which 3/4 of the time comes ups heads one quarter comes up tails then the average is 1/2 all right again doesn't mean that you can measure 1/2 it just means that that's the average value ok a ver egde are important let me give you one observable one kind of observable which is special supposing I take this special class of observables which is one at one state F is one for one state over here and zero for all the others what's the average value of it just a probability right just the probability so if I had an observable which I invented which was one at one place and zero everyplace else then there would be only one term in the sum because F is zero for all the others and for the one case where it's not zero we just get P so if you knew the rules for calculating averages you would also know the rules for calculating for all possible observables you would also a we know the rule for calculating probabilities another way to say it is if you know how to calculate the average value of any observable you can reconstruct for it from it the probability distribution the probabilities for those for that observable probabilities for probabilities for different different for the different possibilities so that raises the question now what is the mathematical representation of observables and that is matrices matrices or linear operators so let's talk again a little bit about matrices and we're like I don't know where I am let's let's talk about observables or or let's talk about matrices but first a matrix is a thing that you can multiply a vector with or act on a vector with it it does something to a vector it's an operation on a vector but it's not just any old operation on the vector it's a linear operation I'm not going to explain that because we won't need to explain it R but whatever it is call it m m for matrix and in the abstract notation it's a thing it's an object which you multiply a vector by and you get a new vector for example one simple example is just multiplying by a complex number multiplying by a complex number is a operator it's a very simple operator you just multiply the vector and you get the amount a vector the number 2 just takes every vector and doubles its size okay all right but the general make the general operators that we're going to consider are ones which are represented by matrices matrices so if we have our vector AE which we represent by a column vector and I'm only going to write two dimensional column vectors because because but you'll immediately deduce what to do if you have a few more entries here all right we can multiply it by a matrix and a matrix as a square array of numbers M 1 1 m12 m21 M 2 2 and in general these numbers are all complex numbers in general they're all complex numbers the A's and the MS in a complex vector space can all be complex numbers and the rule for matrix multiplication which we discussed last time for multiplying a matrix by a vector is just if you want the top entry which you can think of as the top row you take the top row and you multiply it by the vector you take the inner product of the top row with a vector which will be M 1 1 a 1 plus M 1 2 a 2 and then I'm down here and 2 1 a 1 plus M 2 2 a 2 so it's another vector I have to draw a pretty wide because I was adding some numbers here but it's just a vector it's a column vector with one column and it's constructed by taking this row times the column and this row times a column and those are the two entries that's it that's all in matrices are let me give you a couple of examples a couple of simple examples from a very ordinary vector space a very ordinary vector space is just the point there is are the arrows that you can draw in the blackboard you can multiply a vector by in this case a real number you can double it you can triple it you can multiply it by minus 1 and you can add two vectors so vectors that you draw on the blackboard are a vector space let me give you and they have what are their components their components are the X component is X is y and the vector has an X component and a Y component so we represent this vector by its X component that its Y component and we could write it in the form X component of vector Y component of the vector of course now I'm only doing two dimensional space these would be real numbers on the blackboard there wouldn't be complex numbers and this would be a two real dimensional vector space now let's think about some operations the first interesting operation which is the most easy is to stretch the vector space for example multiply every vector by 2 ok multiply of every vector by 2 will sort of magnify the whole space by a factor of 2 any vector will get stretched out to twice its original length that's all what happens here is the matrix that represents that operation it's just the diagonal matrix two two diagonal means that its entries are along the diagonal there are two diagonals the matrix this is called the principal diagonal I think if I remember the other one is called the unprincipled I agonal I don't know what it's called Wow the good diagonal and the bad diagonal this one is the bad diagonal all right let's just calculate what we get all right in the upper entry we get two times VX plus zero so the upper entry just becomes two VX the lower entry we get two times V Y so as advertised this is the operator which simply doubles the length of every vector all right very easy here's more almost similar a similar thing but supposing I just wanted to double the Y component what would that do to the vector space it would take every vector and stretch out its y component by a factor of two without stretching out its X component so it would be a stretch in the Y direction by a factor of two but no stretch in the X direction how would we represent that okay again a diagonal with a 1 here in the X place in the X X and the YY place the fact that - let's check it 1 times V X gives us VX 2 times V y gives us twice V Y so that's a stretching of the vector space in the in the in the Y direction likewise we could put the 2 here and the 1 here that would be a stretching of the vector space which stretches it out in this direction one or two more well it's just a these are these are very easy examples so far they've only involved diagonal matrices let me give you another one the the matrix which corresponds to rotating every vector by 90 degrees take any vector and rotate it by 90 degrees in other words this is the operation which rotates the plane by 90 degrees I'll tell you what it is and then we'll check and see if we can see why it's off diagonal or 1 1 1 -1 excuse me oh let's do this one first let's do what this is before I do the rotation let's do this let's see what this does this takes zero times V X 1 times V Y it interchanges V Y and V X interchanges we want in V X anybody see geometrically what that corresponds to a reflection it's a reflection of the vector space about this diagonal takes every vector and reflect it about that diagonal just flips it about that diagonal that's this matrix here's another matrix where you put a minus 1 down here what does this give this gives V Y minus V X V Y minus V X ok it interchanges the x and y but then throws in a minus sign for one of them that is an operation I'm going to leave it to you to prove because we don't have time but I'm going to tell you what this does it rotates the vector space by 90 degrees Oh does it go the other way it's an easy way to see all this wood but you eat all the starry sky right here every direction he so here rotates rotates every type of a 90 degree the microphone what's that rotating the one rivet around the x and y component s component is going into - right so what I kept pointing in the X directions going into - the vectors for you so which resources they were rotating by 90 degrees in the o clock rotate clockwise yeah that's what I drew here yeah yeah yeah it rotates vector by 90 degrees so you see what matrices correspond to is they correspond to a transformations of the vector space stretching's rotations more complicated kinds of things another example would be a shear shear this is interesting to try to work out the matrix that corresponds to a shear motion let me tell you what a shear motion is or a sheer motion ish a shear motion takes every vector well it it's super boy slides stuff this way okay so it slides you to the right by an amount proportional to how high you are it takes every point and shifts it to the right by an amount proportional to how high you are there's a matrix that describes that motion the forms the plane tilts this axis over is what it does I'm not going to write that one out I'll see if you can find it the next time I'll tell you what the answer is for a for shear motion but in general matrices correspond to transformations special transformations not all transformations are represented as matrices these are the linear transformations to be specific but these are the ones we're going to be interested in and we're going to be interested in a special class of them now this is very abstract and for the moment you will not see why it is just definition but I got to give you some mathematics and then interspersed it with some physics I'm going to give you the mathematics and today and next time I'm going to show you what the point is there's a notion of a hermitian matrix the notion of a hermitian matrix the notion of an hermitian matrix is corresponds to the notion of a real number it's a kind of concept of reality versus imaginary but it's not that doesn't mean that the entries are all real what it means is that if you take take an element of the matrix mije that's and that's some element of the matrix it could be some big matrix I don't know how big it is and it's somewhere over here and then there's m ji m ji is the reflected matrix element if this is m1 to be over here and then m to 1 would be over here M 3 5 might be over here M 5 3 would be over here so interchanging rows and columns is a kind of reflection of the matrix about the diagonal here a hermitian matrix is one that if you reflect it complex conjugates in other words m IJ is equal to m ji star if this matrix was just a number let's just take a case of numbers and I tell you I have a number which is equal to its own complex conjugate what does that tell you about the number that it's real then it has no imaginary part all right now for a matrix that's not what it says it doesn't say that the entries are real but it's a kind of reality property let me give you some examples of matrices one which is hermitian everybody know how to spell hermitian h ER m ay es h un hermit ian heard HP RMIT I ya know you say some people hungry for me hi my name's with eer her mom is a kind of little know each polynomials okay in other words when you flip it you get the complex conjugate here is some example oh oh that says one thing it says that diagonal elements are real because m11 what does it say that says M 1 1 is M star 1 1 if you flip the rows and columns on the diagonal you get the same answer for example I equals 1 J equals 1 M 1 1 equals M 1 1 star so first of all it says the diagonal elements are real real numbers here it's 7 and 3 but then it says that the off diagonal elements are complex conjugates of each other so here's a matrix for example that is not hermitian 4 and to let not hermitian because 2 is not the complex conjugate of 4 all right this is hermitian 4 is the complex conjugate of 4 so if a matrix is real and it's hermitian it's also symmetric but here's another one which is hermitian 4 + I + 4 - I 4 plus I is the complex conjugate of 4 minus I so this is the notion of a hermitian matrix and now I'm going to tell you that we're going to find next time we're going to postulate next time that hermitian matrices are the quantum version of observables a classical version of observables functions of the the state points they're the quantum version of of observable hermitian operator which means of our mission matrix for the moment that's what we're going yeah what you'd like to do is like the get this vector with that matrix and it's a state of the electron system and well the sum of the probabilities is 1 is this enough so that you can get or listen no when you kick it when you kick an electron you want to kick it with a unitary operator mission operator what's that oh I'll tell you this what I did what a unitary operator is one which doesn't change the length of any vector which is what I think what you were asking about right right hermitian operator does not have that property in general so a unitary matrix is one which doesn't change the length and it's it's the result of an operation that you actually do to the electron you kick it you hit it you do something to it I'll tell you very quickly now well no I won't bother telling you because we'll get into it next time matrices or more generally linear linear operators are the basic quantum concept of the observable the thing that you can measure and we're going to then discuss what kind of matrices correspond to the Z component the up-down component of the spin what kind correspond to the X component of the spin what kind correspond to the Y component what kind of observable what kind of matrix corresponds to the component of the spin along some arbitrary direction and so we're going to see then that there's a connection between these vectors matrices and so forth and real directions of space but that's going to take some time it'll take some some little more pieces without all of this there's no way that you can understand honestly what entanglement is or bells in qualities mean what what the basic setup of quantum mechanics is so it's more mathematical of to be able to teach this without all this mathematics but I can't there's no way so you want honestly you want it correctly you want the the basic ingredients that go into quantum mechanics we have to go through this bit of mathematics once we're through it hermitian operators vectors and so forth then the rest is kind of smooth sailing it's it's it's relatively easy stuff but so far we've had to spend more time on mathematics and I might've liked but okay the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Channel: Stanford

Views: 399,237

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Keywords: Science, physics, math, theory, relativity, equation, formula, Leonard, Susskind

Id: VtBRKw1Ab7E

Length: 108min 39sec (6519 seconds)

Published: Wed Apr 23 2008